Modeling of Uncertain Nonlinear System With Z-Numbers

Modeling of Uncertain Nonlinear System With Z-Numbers

Raheleh Jafari (School of Design, University of Leeds, Leeds, UK), Sina Razvarz (Department of Automatic Control, Centre for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV-IPN), Mexico City, Mexico), Alexander Gegov (School of Computing, University of Portsmouth, UK) and Satyam Paul (Department of Engineering Design and Mathematics, University of the West of England, Bristol, UK)
Copyright: © 2021 |Pages: 25
DOI: 10.4018/978-1-7998-3479-3.ch022


In order to model the fuzzy nonlinear systems, fuzzy equations with Z-number coefficients are used in this chapter. The modeling of fuzzy nonlinear systems is to obtain the Z-number coefficients of fuzzy equations. In this work, the neural network approach is used for finding the coefficients of fuzzy equations. Some examples with applications in mechanics are given. The simulation results demonstrate that the proposed neural network is effective for obtaining the Z-number coefficients of fuzzy equations.
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Interpolation technique is extensively implemented in order to function estimation (Boffi & Gastaldi, 2006; Jafarian et al., 2016; Mastylo, 2010). In (Schroeder et al., 1991) a systolic algorithm in order to interpolate as well as evaluate of polynomials by utilizing a linear array of processors is proposed. Two-dimensional polynomial interpolation is described in (Zoli, 2008). In (Neidinger, 2009) a multivariable interpolation method is proposed. In (Olver, 2006), the multivariate Vandermode matrix is implemented. Recently, Smooth function estimation is extensively used (Szabados &Vertesi 1990; Tikhomirov, 1990) which leads a model by using Lagrange interpolating polynomials at the points of product grids (Barthelmann et al., 2000; Xiu & Hesthaven, 2005). However, if there are uncertainties in the interpolation points, none of these techniques will work properly.

The fuzzy equation is taken to be the general form of the fuzzy polynomial. The concept of fuzzy modeling is based on finding the fuzzy coefficients of the fuzzy equation. Various methods have been developed (Goetschel &Voxman, 1986; Kajani et al., 2005; Mamdani, 1976; Mazandarani & Kamyad, 2013; Salahshour et al., 2012; Takagi & Sugeno, 1985; Wang & liu, 2011; Jafari & Yu, 2015a, 2015b, 2015c, 2015d, 2017a; Jafari et al, 2019a, 2016; Razvarz & Jafari, 2017a, 2017b; Razvarz et al., 2017, 2018). In (Friedman et al., 1998) a general fuzzy linear system is studied utilizing the embedding method. In (Buckley & Qu, 1990) the necessary and sufficient conditions for linear as well as quadratic equations are presented in order to contain a solution in a case that the parameters are either real or complex fuzzy numbers. The homotypic analysis technique is discussed in (Abbasbandy, 2006). The Newton's method is proposed in (Abbasbandy & Ezzati, 2006b). In (Allahviranloo et al., 2007), fixed point technique is suggested in order to solve a system of fuzzy nonlinear equations. Nevertheless, these techniques are so much complex.

Iterative technique (Llanas & Sainz, 2006), interpolation technique (Waziri & Majid, 2012) and Runge-Kutta technique (Pederson & Sambandham, 2008) can be applied for obtaining the numerical solution of the fuzzy equation. These techniques can also be applied to fuzzy differential equations (Kajani & Lupulescu, 2009). Neural networks can also be applied in order to find the solutions of the fuzzy equations (Cybenko, 1989; Ito, 2001; Jafari & Razvarz, 2017, 2018, 2015e; Jafari & Yu, 2017b, 2017c; Jafari et al., 2018a, 2018b, 2019b; Jafarian et al., 2016; Jafarian & Jafari, 2012; Yu & Jafari, 2019). In (Buckley & Eslami, 1997), neural network technique is utilized in order to solve the fuzzy quadratic equation. In (Jafari & Razvarz, 2017; Jafarian et al., 2012), the neural network is used in order to obtain the numerical solution of the dual fuzzy equation. In (Mosleh, 2013) neural network method is utilized in order to find the approximate solution of the fully fuzzy matrix equation.

Key Terms in this Chapter

Algorithm: An algorithm in mathematics is a procedure, a description of a set of steps that can be used to solve a mathematical computation.

Nonlinear System: A system in which the change of the output is not proportional to the change of the input.

Neural Network: A neural network is a set of layers (a layer has set of neurons) stacked together sequentially.

Function: A relation between sets that associates to every element of a first set exactly one element of the second set.

Z-Number: Z-number associated with an uncertain variable X as an ordered pair of fuzzy numbers (A,B) where A represents a value of the variable and B represents an idea of certainty.

Polynomial: An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).

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